Since f, ... f(n) are continuous in the environment of a, we can use
Rolle's theorem on g(x) and all the
derivatives.

g(0)=g(h) => there is a c1 between 0 and h such that g'(c1) = 0.
g'(0)=g'(c1) => there is a c2 between 0 and h such that g"(c2) = 0.
...
g(n)(0)=g(n)(cn) =>there is a cn+1 between 0 and h such
that g(n+1)(cn+1) = 0.

And from this last conclusion, we can write

0 = f(n+1)(a+cn+1)-r
<=> f(n+1)(c) = r for a value c between a and a+h.

The term hn+1.f(n+1)(c)/(n+1)! is called
'the derivative form of the remainder'.